So say you’re trying to make a hexaflexagon out of a strip of paper, and you’ve found
that the key to getting a smoothly flexing finished product is to have perfect triangles.
And that reminds you of a trick you learned for folding paper snowflakes.
See, the easy but inaccurate way to fold paper snowflakes has you fold paper in half, then
in half again, and in half again, and then you cut it and unfold it and you’ve got
a snowflakey-looking thing, or you could fold it in half four times to get an even more
snowflakey-looking thing, but real snowflakes have 6-fold symmetry, they’re hexagony-like,
whereas this thing has 8-fold symmetry, which you figure is just what happens when you fold
something in 2, and then in 2, and thenin 2, because each time you’re doubling the
amount of folds and 2*2*2 is 8.
Wait that’s not right…
let’s just put in that extra fold… 8.
Anyway the point is that to get 6-fold symmetry you can fold the paper in half and then in
thirds, which you can do by kind of messing with these two folds until they’re even.
Just like when making a hexaflexagon.
Except that if you stop there you get 3-fold symmetry, you have to fold it in half one
more time to get six fold symmetry.
And that reminds you of an argument you were having on the internet the other day, y’know
the kind where someone says something that you know isn’t true but they’re convinced
they’re right because they have this logic-flavored argument that anyone with actual experience
in the matter knows is completely disconnected from reality, like, yes if you fold a paper
in 2 and then fold that in 2 and then that in 2 again, that has the feel of 2*2*2, and
yes 2*2*2 is 8, but that doesn’t mean it’s illogical to assert that the resulting snowflake
has only 4-fold symmetry and maybe you should put more thought into why that’s true instead
of repeating over and over all the reasons why your theory makes more sense and getting
defensive about the backlash from people who know you have no snowflake-folding experience.
Ahem.
Anyway, the hexagon connection makes you wonder if you can fold and cut a hexaflexagon to
make a hexaflexaflake.
You could think about it first or you could just try it, I mean maybe you learned from
experience that with paper snowflakes if you cut the entire folded edge off then it falls
apart so you figure something similar might happen with hexaflexaflakes only there’s
more folded edges, like with a snowflake you can cut off the entire outside part but if
you try that with a hexaflexagon it will soon be hexaflexa-gone.
But you figure you can probably cut a blip out of the center, and that seems to work
without it falling apart.
But can you still flex it?
Ummmmm… yes!
Except your center seems to have decentralized, but if it worked last time it should work
again with this new center, right?
And now we know that when we flex it this center will become decentralized and the old
center will come back.
Or not.
Both the centers can just hang out on the edges, that’s good too.
Ok so maybe you annotate your centers so you can see that there’s the orange center,
yellow center, and pink center, which makes sense because there’s three different sides
to a hexaflexagon and each side has its own center, the orange center belongs to the orange
side, the yellow center belongs to the yellow side, and finally the pink side with the pink
center, and those are the three sides of a hexaflexagon.
uh… or maybe there’s a pink side with an orange center too, I mean why not, happens
all the time, of course there’s a pink side with pink center and pink side with orange
center and of course there’s a pink side with a yellow center too, somewhere, I mean
it would make sense,
or maybe not?
Let’s just say the pink side with the yellow center is a theoretical possibility that remains
unobserved, as in practice it seems to be… unopenupable.
But anyway when you fold the hexaflexaflake for cutting you notice that all the centers
line up and it doesn't matter which center is the current center so next time you can
cut everything all at once instead of opening up to flex each time.
Of course this theoretical possibility should probably be tested because sometimes sciencing
your math is a good way to check your thought process and inspire your thinking.
And mathematicians often visualize real-world things or shapes or the abstract feel of folding
when they write down notation, like, the thing going on in a mathematician’s head is not
what gets written on paper, just like how people have all sorts of different things
going on in their heads when they say words.
But sometimes you accidentally connect ideas to notation the wrong way, like, you might
write down 2*2*2 and be thinking fold in half, fold in half, fold in half, equals 8-fold
snowflake!
And the more you connect that real-world idea and those abstract thoughts, the more real
it seems that three folds in half makes an 8-fold snowflake, and when someone says you’re
wrong you might be so used to connecting it the old way that it sounds unbelievable that
you could be wrong, especially if you’re in the habit of thinking very highly of your
inexperienced conjectures, so maybe you’re very vocal about your very logical snowflake
theory and why should you have to listen to those “I actually make paper snowflakes”
people anyway, and that’s a recipe for getting your feelings hurt when you don’t understand
why everyone thinks you’re full of baloney.
But you know you’re right, you know 2*2*2=8, and if those “I actually make paper snowflakes”
people insist that 2*2*2=4 you challenge them to prove it to you and if they won’t engage
that’s just because they’re just closed-minded and afraid of discourse.
Which means you’re better than those “I actually make snowflakes” people, which
means you’d feel shame if you were wrong, but if they actually listened or considered
your ideas then they would know you’re right which proves they’re not listening which
makes them extra wrong, and the tiny voice inside you that worries that you are wrong
evokes such feelings of shame especially now that you’ve doubled and tripled down on
how much better you are than them, and that shame hurts which makes you feel like the
actual-snowflake-folders hurt you which makes them even more wrong and you're not saying
everyone who claims to have ever folded a piece of paper in half three times and got
4-fold symmetry is lying, I mean, maybe they folded it an extra time and don’t remember
or maybe they counted wrong or someone switched out their snowflake while they weren’t looking.
You don’t claim to know all the reasons someone might be wrong about snowflake folding,
but you start a forum for baloney snowflake truthers so you can discuss together the many
reasons why actual-snowflake-folders are so wrong; it’s a place for reasonable folks
who agree that folding a piece of paper in half three times should yield an 8-fold snowflake
and you all tell each other constantly how right you are because 2*2*2 is 8, yes, you're
the authority who should be listened to and if they won’t listen you will make them
listen.
Anyway I hope you enjoy this fun holiday craft.
See you next year!